## Mr. Maoudo Faramba BALDE

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- Written by Daouda Niang Diatta

**Maoudo Faramba BALDE** (from Senegal) PhD student at the University Cheikh Anta Diop of Dakar.

**Ph.D Title :** Statistics and Stochastic Modeling: Statistics for Differential Equations**Supervisor :** Khalifa Es-Sebaiy (ENSA de Marrakech, Université Cadi Ayyad, Maroc)**Co-Supervisor :** Papa Ngom (Université Cheikh Anta Diop de Dakar)**Work place:** University Cheikh Anta Diop of Dakar**Related Task(s) :** Task 4

**Description of the subject :** Time-dependent dynamic processes that follow the laws of finance, physics, physiology or biology are usually described by differential systems. For example, stock price dynamics or short-term interest rates can be described using a wide class of financial differential systems. As another example, in biology, pharmacokinetics consists in the study of the evolution of a drug in an organism. It is described through dynamic systems, the human body being assimilated to a set of compartments within which the drug flows. In these contexts, diffusion models described by stochastic differential equations (SDEs) are natural extensions to the corresponding deterministic models (defined by ordinary differential equations, ODEs) to account for time-dependent or serial correlated residual errors and to handle real life variations in model parameters occurring over time. This variability in the model parameters is most often not predictable, not fully understood or too complex to be modeled deterministically. Thus the SDEs consider errors associated with miss pecifications and approximations in the dynamic system. The parametric estimation of such diffusion processes is a key issue. Statistical inference for diffusion type processes satisfying stochastic differential equations driven by Wiener processes has been studied earlier and a comprehensive survey of various methods is given for instance in Kutoyants (2004), Liptser and Shiryaev (2001), Prakasa Rao (2010). Least squares estimator (LSE) is asymptotically equivalent to the MLE. From a practical point of view, in parametric inference, it is more realistic and interesting to consider asymptotic estimation for diffusion processes based on discrete observations. Different methods have been developed in the literature for parametric estimation problems for diffusion processes based on discrete observations by studying consistency and asymptotic normality for the proposed estimators.

The aim of this research project is to extend some cases of diffusion processes driven by Brownian motions to the case of SDEs driven by a fractional Brownian motion (fBm). In this case it is more difficult to work with MLE. This difficulty becomes from the fact of non-existence, in general, of Girsanov theorem for SDEs driven by fBm. For that we propose to use least squares estimators (LSE), which was recently the main tool for studying many problems of parameter estimation for fractional SDEs.

The parametric estimation problems for fractional diffusion processes based on continuous-time observations have been studied e.g. in Kleptsyna and Le Breton (2002), Tudor and Viens (2007), Prakasa Rao (2010) via maximum likelihood method.

From a practical point of view, in parametric inference, it is more realistic and interesting to consider asymptotic estimation for fractional diffusion processes based on discrete observations. In general, the study of the asymptotic distribution of any estimator is not very useful for practical purposes unless the rate of convergence of its distribution is known. The rate of convergence of the distribution of LSE for some diffusion processes driven by Brownian motions based on discrete time data was studied e.g. in Prakasa Rao (2010). In the fractional case we investigated the consistency and the rate of convergence to normality of the LSE for fractional Ornstein-Uhlenbeck process in Es-Sebaiy (2013). Cénac and Es-Sebaiy (2012) studied the ASCLT for least square estimators for Ornstein-Uhlenbeck process driven by fractional Brownian motion.

Conclusion, our aim, using Malliavin calculus, is to study estimation problems for more stochastic models driven by fractional Brownian motion, stochastic partial differential equations (SPDEs) driven by fbm based on discrete and continuous observations, in particular stochastic models applied in Biostatistics and Solar Power

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